A001887 -id:A001887 - cp's OEIS frontend

A080061 Triangle of coefficients of polynomials P(n; x) = Permanent(M), where M=[m(i,j)] is n X n matrix defined by m(i,j)=x if 0<=i-j<=2 else m(i,j)=1.

1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 1, 4, 8, 10, 1, 5, 21, 38, 34, 21, 1, 33, 122, 209, 206, 109, 40, 1, 236, 849, 1400, 1351, 836, 295, 72, 1, 1918, 6719, 10849, 10543, 6629, 2821, 715, 125, 1, 17440, 59873, 95516, 92708, 60284, 26870, 8372, 1604, 212, 1, 175649, 593686

Offset: 0

Vladeta Jovovic, Vladimir Baltic, Jan 23 2003

			1;
0,1;
0,1,1;
0,1,4,1;
1,4,8,10,1;
5,21,38,34,21,1;
... P(5; x) = Permanent(Matrix(5, 5, [[x,1,1,1,1],[x,x,1,1,1],[x,x,x,1,1],[1,x,x,x,1],[1,1,x,x,x]]))= 5+21*x+38*x^2+34*x^3+21*x^4+x^5.

J. Riordan, The enumeration of permutations with three-ply staircase restrictions, unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963. See Table 1. - N. J. A. Sloane, Aug 27 2013 (See A001883)

R. J. Mathar, Table of n, a(n) for n = 0..209

Row sums = A000142, first column = A001887, second column = A001888, third column = A001889, fourth column = A001890, A080018.

Cf. A001883.

A080061_line := proc(n)
    local M,r,c,p,pord ;
    if n = 0 then
        return [1] ;
    else
        M := Matrix(n,n) ;
        for r to n do
        for c to n do
            if r-c >=0 and r-c <=2 then
                M[r,c] := x ;
            else
                M[r,c] := 1 ;
            end if;
        end do:
        end do:
        p := LinearAlgebra[Permanent](M) ;
        pord := degree(p) ;
        [seq( coeff(p,x,r),r=0..pord)] ;
    end if;
end proc:
for n from 0 to 10 do
    print(A080061_line(n)) ;
end do: # R. J. Mathar, Sep 18 2013

M[n_] := Table[If[0 <= i-j <= 2, x, 1], {i, 1, n}, {j, 1, n}]; M[0]={{1}}; Table[CoefficientList[Permanent[M[n]], x], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 06 2016 *)

A078509 Number of permutations p of {1,2,...,n} such that p(i)-i != 1 and p(i)-i != 2 for all i.

Original entry on oeis.org

1, 1, 1, 1, 5, 23, 131, 883, 6859, 60301, 591605, 6405317, 75843233, 974763571, 13512607303, 200949508327, 3190881283415, 53880906258521, 964039575154409, 18217997734199113, 362584510633666621, 7580578211464070863, 166099466140519353035, 3806162403831340850651

Offset: 0

Vladimir Baltic, Vladeta Jovovic, Jan 05 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000255, A055790, A001883, A001887, A075851, A075852.

a:= proc(n) option remember; `if`(n<4, 1,
      (n-1)*a(n-1) +(n-3)*a(n-2) +a(n-3))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 10 2014

a = DifferenceRoot[Function[{y, n}, {-y[n] - n y[n+1] - (n+2) y[n+2] + y[n+3] == 0, y[0] == 1, y[1] == 1, y[2] == 1, y[3] == 1}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

From Vladeta Jovovic, Jul 16 2007: (Start)
G.f.: x/(1+x)*Sum_{n>=0} (n+1)!*(x/(1+x)^2)^n.
a(n) = Sum_{k=1..n} (-1)^(n-k)*k!*binomial(n+k-2,2*k-2). (End)
a(n) ~ exp(-2) * n!. - Vaclav Kotesovec, Aug 25 2014

More terms from Alois P. Heinz, Jan 10 2014

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A080061 Triangle of coefficients of polynomials P(n; x) = Permanent(M), where M=[m(i,j)] is n X n matrix defined by m(i,j)=x if 0<=i-j<=2 else m(i,j)=1.

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